Multiplicity of

periodic solutions for Duffing equations

under nonuniform non-resonance conditions

Author:
Chengwen Wang

Journal:
Proc. Amer. Math. Soc. **126** (1998), 1725-1732

MSC (1991):
Primary 34C25, 34B15

MathSciNet review:
1458269

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper is devoted to the study of multiple -periodic solutions for Duffing equations

under the condition of nonuniform non-resonance related to the positive asymptotic behavior of at the first two eigenvalues and of the periodic BVP on for the linear operator , and the condition on the negative asymptotic behavior of at infinity. The techniques we use are degree theory and the upper and lower solution method.

**[1]**Manuel A. del Pino, Raúl F. Manásevich, and Alejandro Murúa,*On the number of 2𝜋 periodic solutions for 𝑢”+𝑔(𝑢)=𝑠(1+ℎ(𝑡)) using the Poincaré-Birkhoff theorem*, J. Differential Equations**95**(1992), no. 2, 240–258. MR**1165422**, 10.1016/0022-0396(92)90031-H**[2]**Manuel del Pino, Raúl Manásevich, and Alberto Montero,*𝑇-periodic solutions for some second order differential equations with singularities*, Proc. Roy. Soc. Edinburgh Sect. A**120**(1992), no. 3-4, 231–243. MR**1159183**, 10.1017/S030821050003211X**[3]**Pavel Drábek and Sergio Invernizzi,*On the periodic BVP for the forced Duffing equation with jumping nonlinearity*, Nonlinear Anal.**10**(1986), no. 7, 643–650. MR**849954**, 10.1016/0362-546X(86)90124-0**[4]**Robert E. Gaines and Jean L. Mawhin,*Coincidence degree, and nonlinear differential equations*, Lecture Notes in Mathematics, Vol. 568, Springer-Verlag, Berlin-New York, 1977. MR**0637067****[5]**J. Mawhin and J. R. Ward,*Nonuniform nonresonance conditions at the two first eigenvalues for periodic solutions of forced Liénard and Duffing equations*, Rocky Mountain J. Math.**12**(1982), no. 4, 643–654. MR**683859**, 10.1216/RMJ-1982-12-4-643**[6]**Chengwen Wang,*Generalized upper and lower solutions method for the forced Duffing equation*, Proc. Amer. Math. Soc.**125**(1997), no. 2, 397–406. MR**1403119**, 10.1090/S0002-9939-97-03947-6**[7]**CHENGWEN WANG, Multiplicity of periodic solutions for Duffing equation,*Communications on Applied Nonlinear Analysis*(to appear).**[8]**D. HAO AND S.MA, Semi-linear Duffing equations crossing resonance points,*J. Differential Equations*133(1997),*pp*98-116. CMP**97:06****[9]**H. Wang and Y. Li,*Periodic solutions for Duffing equations*, Nonlinear Anal.**24**(1995), no. 7, 961–979. MR**1321737**, 10.1016/0362-546X(94)00114-W**[10]**Nelson Dunford and Jacob T. Schwartz,*Linear Operators. I. General Theory*, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR**0117523**

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Additional Information

**Chengwen Wang**

Affiliation:
Department of Mathematics and Computer Science Rutgers University, Newark, New Jersey 07102

Email:
chengwen@andromeda.rutgers.edu

DOI:
https://doi.org/10.1090/S0002-9939-98-04520-1

Received by editor(s):
November 20, 1996

Communicated by:
Hal L. Smith

Article copyright:
© Copyright 1998
American Mathematical Society